One Small Step For Stability, One Giant Leap For Schwarzschild: Boundedness of Scalar Waves on Schwarzschild Spacetimes

Abstract

This paper gives a detailed proof of the boundedness of the scalar wave equation on a Schwarzschild spacetime using modern energy estimates. The first proof of this statement was given by Kay and Wald in by using Killing vector fields and discrete isometries of the spacetime. However, since then, new techniques to analyze vector fields near the event horizon have been developed by Dafermos and Rodnianski in which has given rise to a new proof strategy. This new proof, using the red shift effect, has been sketched in many lecture notes, but many details are left out. This paper fills in those details and presents the theorem in a self-contained manner.